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1.
J Chem Phys ; 145(21): 211920, 2016 Dec 07.
Article in English | MEDLINE | ID: mdl-28799392

ABSTRACT

The temperature at which the classical critical nucleus size is equal to the average size of the cooperatively rearranging regions (CRR) in a supercooled liquid has been referred to as a "cross-over" temperature. We show, for the first time, using published nucleation rate, viscosity, and thermo-physical data, that the cross-over temperature for the lithium disilicate melt is significantly larger than the temperature of the kinetic spinodal and is equal or close to the temperature corresponding to the maximum in the experimentally observed nucleation rates. We suggest that the abnormal decrease in nucleation rates below the cross-over temperature is most likely because, in this regime, the CRR size controls the critical nucleus size and the nucleation rate. This finding links, for the first time, measured nucleation kinetics to the dynamic heterogeneities in a supercooled liquid.

2.
Sci Total Environ ; 476-477: 485-95, 2014 Apr 01.
Article in English | MEDLINE | ID: mdl-24496022

ABSTRACT

Wintertime TSP samples collected in the two megacities of Xi'an, China and New Delhi, India were analyzed for elements, inorganic ions, carbonaceous species and organic compounds to investigate the differences in chemical compositions and sources of organic aerosols. The current work is the first time comparing the composition of urban organic aerosols from China and India and discussing their sources in a single study. Our results showed that the concentrations of Ca, Fe, Ti, inorganic ions, EC, PAHs and hopanes in Xi'an are 1.3-2.9 times of those in New Delhi, which is ascribed to the higher emissions of dust and coal burning in Xi'an. In contrast, Cl(-), levoglucosan, n-alkanes, fatty alcohols, fatty acids, phthalates and bisphenol A are 0.4-3.0 times higher in New Delhi than in Xi'an, which is attributed to strong emissions from biomass burning and solid waste incineration. PAHs are carcinogenic while phthalates and bisphenol A are endocrine disrupting. Thus, the significant difference in chemical compositions of the above TSP samples may suggest that residents in Xi'an and New Delhi are exposed to environmental hazards that pose different health risks. Lower mass ratios of octadecenoic acid/octadecanoic acid (C18:1/C18:0) and benzo(a)pyrene/benzo(e)pyrene (BaP/BeP) demonstrate that aerosol particles in New Delhi are photochemically more aged. Mass closure reconstructions of the wintertime TSP indicate that crustal material is the most abundant component of ambient particles in Xi'an and New Delhi, accounting for 52% and 48% of the particle masses, respectively, followed by organic matter (24% and 23% in Xi'an and New Delhi, respectively) and secondary inorganic ions (sulfate, nitrate plus ammonium, 16% and 12% in Xi'an and New Delhi, respectively).


Subject(s)
Air Pollutants/analysis , Atmosphere/chemistry , Environmental Monitoring , Aerosols , Air Pollution/statistics & numerical data , China , India , Inorganic Chemicals/analysis , Organic Chemicals/analysis , Particulate Matter/analysis , Polycyclic Aromatic Hydrocarbons/analysis , Trace Elements/analysis
3.
Chem Cent J ; 7(1): 108, 2013 Jun 26.
Article in English | MEDLINE | ID: mdl-23800080

ABSTRACT

BACKGROUND: A conventional gravimetry and electro-gravimetry study has been carried out for the precise and accurate purity determination of lead (Pb) in high purity lead stick and for preparation of reference standard. Reference materials are standards containing a known amount of an analyte and provide a reference value to determine unknown concentrations or to calibrate analytical instruments. A stock solution of approximate 2 kg has been prepared after dissolving approximate 2 g of Pb stick in 5% ultra pure nitric acid. From the stock solution five replicates of approximate 50 g have been taken for determination of purity by each method. The Pb has been determined as PbSO4 by conventional gravimetry, as PbO2 by electro gravimetry. The percentage purity of the metallic Pb was calculated accordingly from PbSO4 and PbO2. RESULTS: On the basis of experimental observations it has been concluded that by conventional gravimetry and electro-gravimetry the purity of Pb was found to be 99.98 ± 0.24 and 99.97 ± 0.27 g/100 g and on the basis of Pb purity the concentration of reference standard solutions were found to be 1000.88 ± 2.44 and 1000.81 ± 2.68 mg kg-1 respectively with 95% confidence level (k = 2). The uncertainty evaluation has also been carried out in Pb determination following EURACHEM/GUM guidelines. The final analytical results quantifying uncertainty fulfills this requirement and gives a measure of the confidence level of the concerned laboratory. CONCLUSIONS: Gravimetry is the most reliable technique in comparison to titremetry and instrumental method and the results of gravimetry are directly traceable to SI unit. Gravimetric analysis, if methods are followed carefully, provides for exceedingly precise analysis. In classical gravimetry the major uncertainties are due to repeatability but in electro-gravimetry several other factors also affect the final results.

4.
Environ Monit Assess ; 184(7): 4553-64, 2012 Jul.
Article in English | MEDLINE | ID: mdl-21956335

ABSTRACT

A field campaign on aerosol chemical properties and trace gases measurements was carried out along the Delhi-Hyderabad-Delhi road corridor (spanning about 3,200 km) in India, during February 1-29, 2004. Aerosol particles were collected on quartz and cellulose filters using high volume (PM(10)) sampler at various locations along the route (i.e., urban, semi-urban, rural, and forest areas) and have been characterized for major cations (Na(+), Ca(2+), Mg(2+), K(+), and NH (4) (+)), anions (Cl(-), NO (3)(-), and SO (4)(2-)), and heavy metals (Cu, Cd, Fe, Zn, Mn, and Pb). Simultaneously, we measured NO(2) and SO(2) gases. These species show large spatial and temporal variations. The ambient PM(10) concentration has been observed to be the highest (55 ± 4 µg m(-3)) near semi-urban areas followed by forest areas (48 ± 2 µg m(-3)) and in rural areas (44 ± 22 µg m(-3)). The concentrations of NO( x ) (NO(2)+NO) and SO(2) ranged from 16 to 69 µg m(-3) and 4 to 11 µg m(-3), respectively. Among anions, NO(3)(-) and SO(4) (2-) are the major constituents of PM(10). The urban and semi-urban sites showed enhanced concentrations of Fe, Zn, Mn, Cd, and Pb. This study provide information about atmospheric concentrations of various species in the northern to central India, which may be important for policy makers to better understand the air quality of the region.


Subject(s)
Aerosols/analysis , Air Pollutants/analysis , Metals, Heavy/analysis , Aerosols/chemistry , Air Pollutants/chemistry , Air Pollution/statistics & numerical data , Atmosphere/chemistry , Environmental Monitoring , India , Metals, Heavy/chemistry , Particulate Matter/analysis , Particulate Matter/chemistry
5.
Chem Cent J ; 5: 17, 2011 Apr 05.
Article in English | MEDLINE | ID: mdl-21466671

ABSTRACT

Arsenic is the toxic element, which creates several problems in human being specially when inhaled through air. So the accurate and precise measurement of arsenic in suspended particulate matter (SPM) is of prime importance as it gives information about the level of toxicity in the environment, and preventive measures could be taken in the effective areas. Quality assurance is equally important in the measurement of arsenic in SPM samples before making any decision. The quality and reliability of the data of such volatile elements depends upon the measurement of uncertainty of each step involved from sampling to analysis. The analytical results quantifying uncertainty gives a measure of the confidence level of the concerned laboratory. So the main objective of this study was to determine arsenic content in SPM samples with uncertainty budget and to find out various potential sources of uncertainty, which affects the results. Keeping these facts, we have selected seven diverse sites of Delhi (National Capital of India) for quantification of arsenic content in SPM samples with uncertainty budget following sampling by HVS to analysis by Atomic Absorption Spectrometer-Hydride Generator (AAS-HG). In the measurement of arsenic in SPM samples so many steps are involved from sampling to final result and we have considered various potential sources of uncertainties. The calculation of uncertainty is based on ISO/IEC17025: 2005 document and EURACHEM guideline. It has been found that the final results mostly depend on the uncertainty in measurement mainly due to repeatability, final volume prepared for analysis, weighing balance and sampling by HVS. After the analysis of data of seven diverse sites of Delhi, it has been concluded that during the period from 31st Jan. 2008 to 7th Feb. 2008 the arsenic concentration varies from 1.44 ± 0.25 to 5.58 ± 0.55 ng/m3 with 95% confidence level (k = 2).

6.
J Environ Monit ; 13(4): 1073-81, 2011 Apr.
Article in English | MEDLINE | ID: mdl-21350781

ABSTRACT

Emission from field burning of agricultural crop residue is a common environmental hazard observed in northern India. It has a significant potential health risk for the rural population due to respirable suspended particulate matter (RSPM). A study on eight stage size segregated mass distribution of RSPM was done for 2 wheat and 3 rice crop seasons. The study was undertaken at rural and agricultural sites of Patiala (India) where the RSPM levels remained close to the National Ambient Air quality standards (NAAQS). Fine particulate matter (PM(2.5)) contributed almost 55% to 64% of the RSPM, showing that, in general, the smaller particles dominated during the whole study period with more contribution during the rice crop as compared to that of wheat crop residue burning. Fine particulate matter content in the total RSPM increased with decrease in temperature. Concentration levels of PM(10) and PM(2.5) were higher during the winter months as compared to that in the summer months. Background concentration levels of PM(10), PM(2.5) and PM(10-2.5) were found to be around 97 ± 21, 57 ± 15 and 40 ± 6 µg m(-3), respectively. The levels increased up to 66, 78 and 71% during rice season and 51, 43 and 61% during wheat crop residue burning, respectively. Extensive statistical analysis of the data was done by using pair t-test. Overall results show that the concentration levels of different size particulate matter are greatly affected by agricultural crop residue burning but the total distribution of the particulate matter remains almost constant.


Subject(s)
Crops, Agricultural , Particle Size , Seasons , India
7.
J Environ Sci Eng ; 52(1): 33-6, 2010 Jan.
Article in English | MEDLINE | ID: mdl-21114104

ABSTRACT

The objective of this study was to monitor the concentration of trace metals in rice. Eight different commercial rice samples were collected from retail market and among these samples Fe, Cd, Cr and Zn metal concentrations were determined by using Flame Atomic Absorption Spectrometer (FAAS) and Inductively Coupled Plasma Atomic Emission Spectrophotometer (ICP-AES). The powdered rice samples were digested by wet chemical method. The analytical results obtained by both the instruments were found comparable. The accuracy of the method has been confirmed by analyzing Certified Reference Material CRM No. 10-b of National Institute for Environmental Studies (NIES). The concentrations of Fe, Cd, Cr and Zn in different brands were found in the range of 15.4 +/- 1.1 to 57.7 +/- 2.5 mg/kg, 0.08 +/- 0.03 to 0.39 +/- 0.05 mg/kg, 0.16 +/- 0.03 to 0.58 +/- 0.08 mg/kg and 7.3 +/- 0.3 to 18.6 +/- 0.5 mg/kg respectively. The standard deviation of the measurements has been calculated for Fe, Cd, Cr and Zn in six replicates of each sample and was found to be less than +/- 3% by the method proposed.


Subject(s)
Food Analysis/methods , Oryza/chemistry , Spectrophotometry, Atomic/methods , Trace Elements/analysis , Cadmium/analysis , Chromium/analysis , Food Contamination/analysis , Humans , India , Iron/analysis , Oryza/toxicity , Trace Elements/toxicity , Zinc/analysis
8.
Sci Total Environ ; 408(20): 4440-5, 2010 Sep 15.
Article in English | MEDLINE | ID: mdl-20637491

ABSTRACT

Variations in pulmonary function tests (PFTs) due to agriculture crop residue burning (ACRB) on children between the age group of 10 to 13 years and the young between 20 to 35 years are studied. The effects of exposure to smoke due to rice-wheat crop residue burning on pulmonary functions like Force Vital Capacity (FVC), Force Expiratory Volume in one second (FEV(1)), Peak Expiratory Flow (PEF) and Force Expiratory Flow in 25 to 75% of FVC (FEF(25-75%)) on 40 healthy subjects of rural/agricultural area of Sidhuwal village of Patiala City were investigated for a period from August 2008 to July 2009. Measurements were taken by spirometry according to the American Thoracic Society standards. High volume sampler (HVS) and Anderson Impactor were used to measure the concentration levels of SPM, PM(10) and PM(2.5) in ambient air of the Sidhuwal village. A significant increase in the concentration levels of SPM, PM(10) and PM(2.5) was observed due to which PFTs of the subjects showed a significant decrease in their values, more prominently in the case of children. PFTs of young subjects recovered up to some extent after the completion of burning period but the PFT values of children remained significantly lower (p<0.001) even after the completion of burning episodes. Small size particulate matter (PM(2.5) and PM(10)) affected the PFTs to a large extent in comparison to the large size particulate matter (SPM). The study indicates that ACRB is a serious environmental health hazard and children are more sensitive to air pollution, as ACRB poses some unrecoverable influence on their PFTs.


Subject(s)
Air Pollutants/toxicity , Crops, Agricultural , Incineration , Respiratory Function Tests , Adolescent , Adult , Air Pollutants/analysis , Atmosphere/chemistry , Child , Environmental Exposure/analysis , Environmental Monitoring , Female , Humans , India , Lung/drug effects , Male , Particulate Matter/analysis , Particulate Matter/toxicity , Young Adult
9.
Environ Monit Assess ; 169(1-4): 1-13, 2010 Oct.
Article in English | MEDLINE | ID: mdl-19757121

ABSTRACT

The variation in air quality was assessed from the ambient concentrations of various air pollutants [total suspended particle (TSP), particulate matter < or =10 microm (PM(10)), SO(2), and NO(2)] for pre-Diwali, Diwali festival, post-Diwali, and foggy day (October, November, and December), Delhi (India), from 2002 to 2007. The extensive use of fireworks was found to be related to short-term variation in air quality. During the festival, TSP is almost of the same order as compared to the concentration at an industrial site in Delhi in all the years. However, the concentrations of PM(10), SO(2), and NO(2) increased two to six times during the Diwali period when compared to the data reported for an industrial site. Similar trend was observed when the concentrations of pollutants were compared with values obtained for a typical foggy day each year in December. The levels of these pollutants observed during Diwali were found to be higher due to adverse meteorological conditions, i.e., decrease in 24 h average mixing height, temperature, and wind speed. The trend analysis shows that TSP, PM(10), NO(2), and SO(2) concentration increased just before Diwali and reached to a maximum concentration on the day of the festival. The values gradually decreased after the festival. On Diwali day, 24-h values for TSP and PM(10) in all the years from 2002 to 2007 and for NO(2) in 2004 and 2007 were found to be higher than prescribed limits of National Ambient Air Quality Standards and exceptionally high (3.6 times) for PM(10) in 2007. These results indicate that fireworks during the Diwali festival affected the ambient air quality adversely due to emission and accumulation of TSP, PM(10), SO(2), and NO(2).


Subject(s)
Air Pollutants/analysis , Environmental Monitoring , Nitrogen Dioxide/analysis , Particulate Matter/analysis , Sulfur Dioxide/analysis , Air Pollution/statistics & numerical data , Atmosphere/chemistry , Holidays , Humans , India , Inhalation Exposure/analysis , Inhalation Exposure/statistics & numerical data , Time , Weather
10.
Proc Natl Acad Sci U S A ; 106(47): 19780-4, 2009 Nov 24.
Article in English | MEDLINE | ID: mdl-19903878

ABSTRACT

The low-temperature dynamics of ultraviscous liquids hold the key to understanding the nature of glass transition and relaxation phenomena, including the potential existence of an ideal thermodynamic glass transition. Unfortunately, existing viscosity models, such as the Vogel-Fulcher-Tammann (VFT) and Avramov-Milchev (AM) equations, exhibit systematic error when extrapolating to low temperatures. We present a model offering an improved description of the viscosity-temperature relationship for both inorganic and organic liquids using the same number of parameters as VFT and AM. The model has a clear physical foundation based on the temperature dependence of configurational entropy, and it offers an accurate prediction of low-temperature isokoms without any singularity at finite temperature. Our results cast doubt on the existence of a Kauzmann entropy catastrophe and associated ideal glass transition.


Subject(s)
Glass/chemistry , Models, Theoretical , Solutions/chemistry , Mathematics , Temperature , Viscosity
11.
J Chem Phys ; 130(23): 234503, 2009 Jun 21.
Article in English | MEDLINE | ID: mdl-19548735

ABSTRACT

Glass transition temperature and fragility are two important properties derived from the temperature dependence of the shear viscosity of glass-forming melts. While direct calculation of these properties from atomistic simulations is currently infeasible, we have developed a new topological modeling approach that enables accurate prediction of the scaling of both glass transition temperature and fragility with composition. A key feature of our approach is the incorporation of temperature-dependent constraints that become rigid as a liquid is cooled. Using this approach, we derive analytical expressions for the composition (x) dependence of glass transition temperature, T(g)(x), and fragility, m(x), in binary alkali borate systems. Results for sodium borate and lithium borate systems are in agreement with published values of T(g)(x) and m(x). Our modeling approach reveals a natural explanation for the presence of the constant T(g) regime observed in alkali borate systems.

12.
J Chem Phys ; 130(9): 094503, 2009 Mar 07.
Article in English | MEDLINE | ID: mdl-19275405

ABSTRACT

We present a topological model for the composition dependence of glass transition temperature and fragility. Whereas previous topological models are derived for zero temperature conditions, our approach incorporates the concept of temperature-dependent constraints that freeze in as the system is cooled from high temperature. Combining this notion of temperature-dependent constraints with the Adam-Gibbs model of viscosity, we derive an analytical expression for the scaling of glass transition temperature and fragility in the binary Ge(x)Se(1-x) system. In the range of 0

13.
Environ Monit Assess ; 156(1-4): 407-18, 2009 Sep.
Article in English | MEDLINE | ID: mdl-18720020

ABSTRACT

Chemical measurements often constitute the basis for informed decision-making at different levels in society; sound decision-making is possible only if the quality of the data used is uncompromised. To guarantee the reliability and comparability of analytical data an intricate system of quality-assurance measures has to be put into effect in a laboratory. Reference materials and, in particular, certified reference materials (CRMs) are essential for achieving traceability and comparability of measurement results between laboratories and over time. As in any other domain of analytical chemistry, techniques used to monitor the levels and fate of contaminants in the environment must be calibrated using appropriate calibration materials, and the methods must be properly validated using fit-for purpose matrix-matched CRMs, to ensure confidence in the data produced. A sufficiently large number of matrix CRMs are available for analysis of most elements, and the group of chemicals known as persistent organic pollutants, in environmental compartments and biota. The wide variety of analyte/level/matrix/matrix property combinations available from several suppliers enables analysts to select CRMs which sufficiently match the properties of the samples they analyse routinely. Certified reference materials are playing an increasing role in the monitoring of environmental pollution. This paper is an attempt for describing the recent development of certified reference materials for road transportation, which mainly covers the combustion of fossil fuel, road side dust etc.


Subject(s)
Environmental Monitoring/standards , Reference Standards , Transportation , Models, Theoretical
14.
Chemosphere ; 74(4): 590-8, 2009 Jan.
Article in English | MEDLINE | ID: mdl-18996564

ABSTRACT

A state-wise assessment of methane (CH(4)) budget for Indian paddies, based on a decadal measurement data across India is presented for the calendar year (CY) 1994, the base year for India's Initial National Communication (NATCOM) to the United Nations Framework Convention on Climate Change (UNFCCC), along with national trend from CY 1979 to 2006. The NATCOM CH(4) emission factors (EFs) for Indian paddy cultivation areas, generally having less than 0.7% of soil organic carbon (SOC), have been estimated as 17.48+/-4 g m(-2) for irrigated continuously flooded (IR-CF), 6.95+/-1.86 g m(-2) for rain-fed drought prone (RF-DP), 19+/-6 g m(-2) for rain-fed flood prone (RF-FP) and deep-water (DW), 6.62+/-1.89 g m(-2) for irrigated intermittently flooded single aeration (IR-IF-SA) and 2.01+/-1.49 g m(-2) for IR-IF multiple aeration (MA) paddy water regimes. The state-wise study for 1994 has indicated national CH(4) budget estimate of 4.09+/-1.19 Tg y(-1) and the trend from 1979 to 2006 was in the range of 3.62+/-1 to 4.09+/-1.19 Tg y(-1). Four higher emitting or "hot spot" states (West Bengal, Bihar, Madhya Pradesh and Uttar Pradesh) have accounted for 53.9% of total CH(4) emission with RF-FP paddy water regime as the major contributor. CH(4) emissions were enhanced by factors such as SOC ( approximately 1.5 times due to increase in SOC by approximately 1.8 times), paddy cultivars (approximately 1.5 times), age of seedlings (approximately 1.4 times), and seasons (approximately 1.8 times in Kharif or monsoon than in Rabi or winter season).


Subject(s)
Agriculture , Methane/analysis , Carbon/analysis , India , Oryza , Seasons , Soil
16.
Chemosphere ; 71(4): 750-8, 2008 Mar.
Article in English | MEDLINE | ID: mdl-18068211

ABSTRACT

Municipal solid waste generation rate is over-riding the population growth rate in all mega-cities in India. Greenhouse gas emission inventory from landfills of Chennai has been generated by measuring the site specific emission factors in conjunction with relevant activity data as well as using the IPCC methodologies for CH4 inventory preparation. In Chennai, emission flux ranged from 1.0 to 23.5mg CH4m(-2)h(-1), 6 to 460microg N2Om(-2)h(-1) and 39 to 906mg CO2m(2)h(-1) at Kodungaiyur and 0.9 to 433mg CH4m(-2)h(-1), 2.7 to 1200microg N2Om(-2)h(-1) and 12.3 to 964.4mg CO2m(-2)h(-1) at Perungudi. CH4 emission estimates were found to be about 0.12Gg in Chennai from municipal solid waste management for the year 2000 which is lower than the value computed using IPCC, 1996 [IPCC, 1996. Report of the 12th session of the intergovernmental panel of climate change, Mexico City, 1996] methodologies.


Subject(s)
Cities , Gases/analysis , Greenhouse Effect , Refuse Disposal , Carbon Dioxide/analysis , Gases/chemistry , India , Methane/analysis , Nitrous Oxide/analysis , Organizations
17.
Phys Rev E Stat Nonlin Soft Matter Phys ; 78(6 Pt 1): 062501, 2008 Dec.
Article in English | MEDLINE | ID: mdl-19256888

ABSTRACT

We derive an analytical expression showing that the fragility of a supercooled liquid is a result of (i) a thermodynamic term depending on change in configurational entropy and (ii) a kinetic term depending on change in the nonexponentiality or "stretching" of the relaxation function, as quantified by the exponent beta of the Kohlrausch-Williams-Watts (KWW) relaxation function. Our expression indicates that there is not a direct correlation between the non-Arrhenius scaling of liquid viscosity and the nonexponential nature of glassy relaxation. Rather, the temperature dependence of the stretching exponent beta provides a lower limit for fragility, which can be increased through changes in the configurational entropy. Our result explains the apparent contradiction between those researchers showing a correlation between beta and fragility and those who question such a correlation due to the spread of the data.

18.
J Phys Chem A ; 111(32): 7957-65, 2007 Aug 16.
Article in English | MEDLINE | ID: mdl-17649986

ABSTRACT

We propose a technique for computing the master equation dynamics of systems with broken ergodicity. The technique involves a partitioning of the system into components, or metabasins, where the relaxation times within a metabasin are short compared to an observation time scale. In this manner, equilibrium statistical mechanics is assumed within each metabasin, and the intermetabasin dynamics are computed using a reduced set of master equations. The number of metabasins depends upon both the temperature of the system and its derivative with respect to time. With this technique, the integration time step of the master equations is governed by the observation time scale rather than the fastest transition time between basins. We illustrate the technique using a simple model landscape with seven basins and show validation against direct Euler integration. Finally, we demonstrate the use of the technique for a realistic glass-forming system (viz., selenium) where direct Euler integration is not computationally feasible.

19.
J Chem Phys ; 126(22): 224504, 2007 Jun 14.
Article in English | MEDLINE | ID: mdl-17581060

ABSTRACT

The phenomenology of the laboratory glass transition is examined in the enthalpy landscape framework. It is shown that a generic description of the glassy state based on partitioning of the phase space caused by the finiteness of the time of observation explains all universal features of glass transition. Using this description of glass, which is referred to as the extrinsically constrained liquid, expressions are derived for properties of glass and for property changes at the laboratory glass transition. A model enthalpy landscape is used to illustrate the basic concepts of this description. Additional new consequences of this description, such as the role of complexity in glass transition and the zero residual entropy of a glass, are discussed.

20.
J Chem Phys ; 126(18): 184511, 2007 May 14.
Article in English | MEDLINE | ID: mdl-17508815

ABSTRACT

A system that is initially ergodic can become nonergodic, i.e., display "broken ergodicity," if the relaxation time scale of the system becomes longer than the observation time over which properties are measured. The phenomenon of broken ergodicity is of vital importance to the study of many condensed matter systems. While previous modeling efforts have focused on systems with a sudden, discontinuous loss of ergodicity, they cannot be applied to study a gradual transition between ergodic and nonergodic behavior. This transition range, where the observation time scale is comparable to that of the structural relaxation process, is especially pertinent for the study of glass transition range behavior, as ergodicity breaking is an inherently continuous process for normal laboratory glass formation. In this paper, we present a general statistical mechanical framework for modeling systems with continuously broken ergodicity. Our approach enables the direct computation of entropy loss upon ergodicity breaking, accounting for actual transition rates between microstates and observation over a specified time interval. In contrast to previous modeling efforts for discontinuously broken ergodicity, we make no assumptions about phase space partitioning or confinement. We present a hierarchical master equation technique for implementing our approach and apply it to two simple one-dimensional landscapes. Finally, we demonstrate the compliance of our approach with the second and third laws of thermodynamics.

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